We fabricate a microelectromechanical systems (MEMS) microphone that incorporates a graphene-based membrane that vibrates in response to acoustic forcing. We employ a novel fabrication process, where a graphene/poly(methyl methacrylate) (PMMA) bilayer membrane is transferred over a cavity on a separate chip before being affixed to the surface of another chip containing an electrode, resulting in the fabrication of a moveable capacitor with a membrane-to-electrode gap of 8 $\mu \text{m}$ . The gap, which is less than half the size of other reported graphene membrane-based audio transducers, allows for the device to operate with the low-DC bias voltages of about 1 V and, when integrated with a custom-designed readout circuit, demonstrates a sensitivity to sound pressure between 0.1 and 10 mV/Pa across the range 100 Hz–20 kHz. As well as a sensitivity that is comparable with the previous work, the flat frequency response is stable when the sound pressure is varied between 70 and 80 dBSPL, with the sensitivity value not varying by more than 0.2 mV/Pa.